Theorem breq12 4098

Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)

Assertion

Ref	Expression
breq12	|- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Proof of Theorem breq12

Step	Hyp	Ref	Expression
1		breq1 4096	. 2 |- ( A = B -> ( A R C <-> B R C ) )
2		breq2 4097	. 2 |- ( C = D -> ( B R C <-> B R D ) )
3	1, 2	sylan9bb 462	1 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   class class class wbr 4093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094

This theorem is referenced by:  breq12i  4102  breq12d  4106  breqan12d  4109  posng  4804  isopolem  5973  poxp  6406  rbropapd  6451  ecopover  6845  ecopoverg  6848  ltdcnq  7677  recexpr  7918  ltresr  8119  reapval  8815  ltxr  10071  xrltnr  10075  xrltnsym  10089  xrlttr  10091  xrltso  10092  xrlttri3  10093  xposdif  10178  f1olecpbl  13476  wlk2f  16292  exmidsbthrlem  16750